Superconvergence and asymptotic expansions for linear finite element approximations on criss-cross mesh

In this paper, we discuss the error estimation of the linear finite element solution on criss-cross mesh. Using space orthogonal decomposition techniques, we obtain an asymptotic expansion and superconvergence results of the finite element solution. We first prove that the asymptotic expansion has different forms on the two kinds of nodes and then derive a high accuracy combination formula of the approximate derivatives.     

THE MECHANICAL QUADRATURE METHODS AND THEIR EXTRAPOLATION FOR SOLVING BIE OF STEKLOV EIGENVALUE PROBLEMS

By means of the potential theory Steklov eigenvalue problems are transformed into general eigenvalue problems of boundary integral equations (BIE) with the logarithmic singularity.Using the quadrature rules, the paper presents quadrature methods for BIE of Steklov eigenvalue problem, which possess high accuracies O(h^3) and low computing complexities. Moreover, an asymptotic expansion of the errors with odd powers is shown. Using h^3-Richardson extrapolation, we can not only improve the accuracy order of approximations, but also derive a posterior estimate as adaptive algorithms. The efficiency of the algorithm is illustrated by some examples.     

ARCH(0,1)系数中位无偏估计分布的渐近展开(英文)

This paper is concerned with the distributional properties of a median unbiased estimator of ARCH(0,1)coefficient.The exact distribution of the estimator can be easily derived,however its practical calculations are too heavy to implement, even though the middle range of sample sizes.Since the estimator is shown to have asymptotic normality,asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements.Accuracies of expansion formulas are evaluated numerically,and the results of which show that we can effectively use the expansion as a fine approximation of the distribution with rapid calculations.Derived expansion are applied to testing hypothesis of stationarity,and an implementation for a real data set is illustrated.     

Extrapolation methods to compute hypersingular integral in boundary element methods

The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/sin 2(x-s) is discussed,and the main part of the asymptotic expansion of error function is obtained.Based on the main part of the asymptotic expansion,a series is constructed to approach the singular point.An extrapolation algorithm is presented and the convergence rate is proved.Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

一类拟线性奇摄动方程的边值问题

In this paper, the Nagumo theorem and the fixed-point theorem are used to prove the existence and the uniqueness and to estimate the asymptotic expansion of the shock solutions of the boundary value problems for a class of quasilinear differential equations, the asymptotic expansion of solution of any orders including boundary is obtained.     

Asymptotic expansions of finite element solutions to Robin problems in H 3 and their application in extrapolation cascadic multigrid method

For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.     

Singularly perturbed boundary value problems for a class of second order turning point on infinite interval

This article solved the asymptotic solution of a singularly perturbed boundary value problem with second order turning point, encountered in the dissipative equilibrium vector field of the coupled convection disturbance kinetic equations under the constrained filed and the gravity. Using the matching of asymptotic expansions, the formal asymptotic solution is constructed. By using the theory of differential inequality the uniform validity of the asymptotic expansion for the solution is proved.     

ARCH(0,1)系数中位无偏估计分布的渐近展开

This paper is concerned with the distributional properties of a median unbiased estimator of ARCH（0,1） coefficient. The exact distribution of the estimator can be easily derived, however its practical calculations are too heavy to implement, even though the middle range of sample sizes. Since the estimator is shown to have asymptotic normality, asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements. Accuracies of expansion formulas are evaluated numerically, and the results of which show that we can effectively use the expansion as a fine approximation of the distribution with rapid calculations. Derived expansion are applied to testing hypothesis of stationarity, and an implementation for a real data set is illustrated.     

Existence of Canards under Non-generic Conditions

The canard phenomenon occurring in planar fast-slow systems under nongeneric conditions is investigated. When the critical manifold has a non-generic fold point by using the method of asymptotic analysis combined with the recently developed blow-up technique, the existence of a canard is established and the asymptotic expansion of the parameter for which a canard exists is obtained.     

Instructions for Authors

<正>Submission Authors must use LaTeX for typewriting,and visit our website www.actamath.com to submit your paper.Our address is Editorial Office of Acta Mathematica Sinica,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,P.R.China.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

正August 10-14,2015Beijin,China The International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

FRACTIONAL INTEGRALS OF THE WEIERSTRASS FUNCTIONS： THE EXACT BOX DIMENSION

The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The ezact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.     

English Series

正1 Aims and Scope Acta Mathematicae Applicatae Sinica(English Series)is a quarterly journal established by the Chinese Mathematical Society.The journal publishes high quality research papers from all branches of applied mathematics,particularly welcomes those from partial differential equations,computational mathematics,applied probability,mathematical finance,statistics,dynamical systems,optimization and management science.     

CHINESE ANNALS OF MATHEMATICS SERIES B Volume 35 · 2014 TOTAL CONTENTS

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Congruences,ideals and annihilators in standard QBCC-algebras

We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels. Work on the paper was supported by Council of Czech Government No J14/98:153100011.     

A decomposition of continuity via ideals

A new class of sets in ideal topological spaces is introduced and using these sets, a decomposition of continuity is given.      

When a compact (countably compact) set is closed. II

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T ₄,-spaces.     

ASYMPTOTIC BEHAVIOR OF ECKHOFF＇S METHOD FOR FOURIER SERIES CONVERGENCE ACCELERATION

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called ＂jumps＂ is studied and asymptotic L2 constants of the rate of convergence of the method are computed.